automatic weights won't connect skeleton to mesh
You need to:

Select all you mesh parts
Select your armature while selecting the mesh using Shift+RightClick
Hit Crtl+P then choose With Automatic Weighting

And all will be good as shown below:

As @lemon mentioned, although it will work without joining the mesh parts together, but it's better to join the mesh parts together then apply the parenting to use the dog as one object instead of having many objects representing one object as this will avoid a lot of mess. To join the mesh parts select all mesh parts using Shift+RightClick then hit Ctrl+J